KdV waves in atomic chains with nonlocal interactions

TL;DR

This paper proves the existence of near-sonic solitary waves in atomic chains with nonlocal interactions, extending previous results to a broader class of systems using a novel analytical approach.

Contribution

It introduces a new method employing the contraction mapping principle to analyze nonlocal interactions, broadening the class of atomic chains where KdV waves are proven to exist.

Findings

Existence of near-sonic solitary waves in nonlocal atomic chains

Extension of KdV limit analysis to broader atomic systems

Use of contraction mapping principle instead of Fourier analysis

Abstract

We consider atomic chains with nonlocal particle interactions and prove the existence of near-sonic solitary waves. Both our result and the general proof strategy are reminiscent of the seminal paper by Friesecke and Pego on the KdV limit of chains with nearest neighbor interactions but differ in the following two aspects: First, we allow for a wider class of atomic systems and must hence replace the distance profile by the velocity profile. Second, in the asymptotic analysis we avoid a detailed Fourier pole characterization of the nonlocal integral operators and employ the contraction mapping principle to solve the final fixed point problem.